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The Fokker-Planck (FP) equation is a linear partial differential equationwhich governs the temporal and spatial evolution of the probability densityfunction (PDF) associated with the response of stochastic dynamical systems. Anexact analytical solution of the FP equation is only available for a limitedsubset of dynamical systems. Semi-analytical methods are available for larger,yet still a small subset of systems, while traditional computational methods;e.g. Finite Elements and Finite Difference require dividing the computationaldomain into a grid of discrete points, which incurs significant computationalcosts for high-dimensional systems. Physics-informed learning offers apotentially powerful alternative to traditional computational schemes. Toevaluate its potential, we present a data-free, physics-informed neural network(PINN) framework to solve the FP equation for a class of nonlinear stochasticdynamical systems. In particular, through several examples concerning thestochastic response of the Duffing, Van der Pol, and the Duffing-Van der Poloscillators, we assess the ability and accuracy of the PINN framework in $i)$predicting the PDF under the combined effect of additive and multiplicativenoise, $ii)$ capturing P-bifurcations of the PDF, and $iii)$ effectivelytreating high-dimensional systems. Through comparisons with Monte-Carlosimulations and the available literature, we show that PINN can effectivelyaddress all of the afore-described points. We also demonstrate that thecomputational time associated with the PINN solution can be substantiallyreduced by using transfer learning.

Physics-Informed Solution of The Stationary Fokker-Plank Equation for a Class of Nonlinear Dynamical Systems: An Evaluation Study